Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

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Introduction

Significant progress has been achieved in the analysis of the motion of incom- pressible fluid models of differential type using finite element methods. This work is concerned with the finite element approximation of the boundary value prob- lems for the motion of incompressible fluid governed by the Stokes/Navier-Stokes equations, or by the non-Newtonian Stokes equation with certain nonlinear slip boundary conditions. Since these classes of nonlinear slip boundary conditions include the subdifferential property, the variational formulations are variational inequality problems. So far extensive study has been done for the motion of incompressible fluid which is governed by the Stokes/Navier-Stokes equation, or by the non-Newtonian Stokes/Navier-Stokes equation in hydrodynamics as well as in mathematics. As to the boundary condition, almost all of these works have dealt with the adhe- sive boundary condition to the surface of a rigid body, namely, with the Dirichlet boundary condition (see, for instance, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]). This is of course reasonable from or consistent with the nature of such fluids and walls. However, there are phenomena, whose mathematical analysis seems to require in- troduction of some non-routine boundary conditions which might allow non-trivial motion of fluid on or across the boundary, for instance, slip or leak of fluid at the boundary. As examples, we can refer to flow through a drain or canal with its bottom covered by sherbet of mud and pebbles, flow of melted iron coming out from a smelting furnace, avalanche of water and rocks, blood flow in a vein of an arterial sclerosis patient, flow through a net or sieve, water flow in purification plant, etc. This observation is consistent with the hypothesis that the velocity at the wall is not zero. Several studies have been made and showed not only that slip takes place when a threshold is reached [12] but also it’s the origin of many defects and instabilities in the polymer injection process [13, 14]. The inadequacy of the adherence condition is also evident from experimental ob- servations (e.g.[15, 16, 17]) which show that non-Newtonian fluids such as polymer melts often exhibit macroscopic wall slip, and that in general this is governed by a nonlinear and nonmonotone relation between the slip velocity and the traction. This may be an important factor in sharkskin, spurt and hysteresis effects; see [18, 19, 20] for a detailed discussion and additional references. Moreover, fluids that exhibit boundary slip have important technological applications. For exam- ple, the polishing of artificial heart valves and internal cavities in a variety of manufactured parts is achieved by imbedding such fluids with abrasives [21]. A more important class of slip laws are those in which the magnitude of the tangential stress must reach some critical value, here called the slip yield stress, before slip occurs. These problems are especially interesting because the part of the boundary where slip occurs is not known and may vary with time. In fact, some experiments show that the onset of slip and the slip velocity may also depend on the normal stress at the boundary [15, 17, 22]. Not surprisingly, the theory and the numerical analysis for flow problems of this kind is equally limited. But since the last two decades, a remarkable progress has been achieved in the field of computational fluid dynamics with slip boundary conditions. For a stationary Stokes problem, Fujita [23] introduced the following slip law. |(σn)τ | ≤ g, |(σn)τ | < g ⇒ uτ = wτ , |(σn)τ | = g ⇒ uτ 6= wτ , − (σn)τ = g (u − w)τ |(u − w)τ |    on S, (0.1) where the notations will be explained later. Fujita et al. [23, 24] have studied the existence of steady solution to the Stokes problem with slip condition (0.1) which they call slip of the “friction type”, and with an analogous leak condition and later on Li et al. [25] for Navier-Stokes equations with the slip condition (0.1) . The regularity and the solvability of the solution for Stokes and Navier-Stokes equations have been carried out in [26, 27, 28, 29, 30] and for non-Newtonian Stokes equations by [31]. Regard- ing the numerical analysis for Stokes and Navier-Stokes equations with this slip conditions in terms of finite element method, [32] proposed an iterative algorithm of Uzawa type and gave some numerical examples. Recently Kashiwabara [33] presented the framework of finite element method including existence, unique- ness, error analysis and implementation. Error estimates for the Stokes problem with such slip are obtained in [34]. Based on the penalty method, in [35, 36], Li et al. proposed a finite element approximation combined with penalty method and error estimates with strong regularity assumption on the velocity. Low-order finite elements, such as the P1/P1 element with stabilized terms, are applied in [37, 38, 39]. Another approach by the P1 + /P1 element, based on a saddle-point formulation is found in [40]. In our knowledge, no work has been done in finite element methods regarding the non-Newtonian Stokes equations with slip bound- ary condition (0.1). C. Leroux [41] introduced the following slip boundary condition in Stokes equa- tions and later on in Navier-Stokes equations with A. Tani [42] where they have studied the wellposedness of the steady solution. |(σn)τ | ≤ g, |(σn)τ | < g ⇒ uτ = wτ , |(σn)τ | = g ⇒ uτ 6= wτ , − (σn)τ = (g + h(|(u − w)τ |)) (u − w)τ |(u − w)τ |    on S. (0.2) The threshold slip boundary condition (0.2) arise in the modeling of flows of poly- mer melts during extrusion (where the slip threshold may depend on the normal stress at the boundary) and flows of yield-stress fluids [43, 44, 45]. Regarding the numerical analysis, in our knowledge, there is no work with this slip condition in Stokes and Navier-Stokes equations or in Non-newtonian Stokes equations dealing with finite element methods.

Thesis overview and our contributions

The thesis is divided into three main research chapters. Each of these chapters represents scientific contributions (in form of published, accepted or submitted journal papers). As such, each chapter is intended to be self-contained and can be read independently of the other chapters. Note also that the notation in each chapter is therefore slightly different. Chapter 1 is devoted to the study of finite element analysis for Stokes and Navier-Stokes equations driven by threshold slip boundary conditions of type (0.2) defined in [41]. The principal goal is to analyze from the numerical analysis viewpoint the solvability, stability and convergence of the resulting variational inequalities of such problems. In this chapter, after re-writing the problems in the form of variational inequalities, a fixed point strategy is used to show existence of solutions. The finite element formulation for both Stokes and Navier-Stokes equa- tions are derived and we establish the convergence of the finite element solutions to the continuous solutions of each problems. For Stokes, we consider a scheme related to the variational formulation of second kind and for Navier-Stokes, we consider a scheme related to the Oseen problem and show that their solution re- spectively converges to the finite element solution of the Stokes and Navier-Stokes equations. We formulate and show the convergence of the Uzawa’s algorithm and finally, present some numerical experiments to verify the feasibility of our algo- rithm. This chapter has been the object of the papers [46, 47] and is the first work on finite element approximation dealing with slip boundary con- dition of type (0.2). Chapter 2 is dealing with finite element approximation of the stationary power-law Stokes equations driven by boundary conditions of “friction type” (0.1). The theoretical analysis of this chapter is based on the paper of Han and Reddy [48], where sufficient conditions for existence and uniqueness are derived for the kind of weak formulations we analyzed here. It is shown that by applying a variant approximation formulated is achieved with classical assumptions on the regularity of the weak solution. We also present the implementation of the nonlinear saddle point problem formulated by adopting a particular algorithm based on vanishing viscosity approach and long time behavior of an initial value problem. Finally, the predictions observed by the theory developed are validated by numerical ex- periments presented. This chapter has been the object of the paper [49] and is also the first work on finite element approximation for power-law Stokes equations with slip boundary conditions of “friction type”. Chapter 3 reports on the long-time stability of the Crank-Nicolson scheme for the 2D Navier-Stokes equations driven by the slip boundary conditions of “friction type” (0.1). In the case of the Navier-Stokes equations with Dirichlet boundary conditions, establishing the H1 -stability for all time using Crank-Nicolson scheme on time has be proven in [50]. Our object here is to extend the results to the case of slip boundary condition. We discretize these equations using the Crank- Nicholson scheme in time and in space the finite element approximation. We establish its well-posedness and stability of the numerical scheme on L 2 -norm and H1 -norm for all positive time. This chapter has been the object of the paper [51]. Over the past few years a remarkable progress has been achieved in the field of computational contact mechanics. One of the key ingredients in this phenomenal growth is attributed to the better mathematical understanding of problems. The formulation by means of variational inequalities (see [69, 71, 57, 58, 72, 73, 74]) and the finite element method have contributed to the development of reliable frameworks for the numerical treatment of such problems. Despite such advances in the modeling and numerical treatment of contact problems with friction, it should be mentioned that most works reported in the literature are still restricted to solid mechanics. The numerical analysis works dealing with fluids flow are concerned with the standard Amontons-Coulomb law of perfect friction [75, 76, 49, 35, 77, 37, 78, 40, 33], replacing (1.5) by |(σn)τ | ≤ g, |(σn)τ | < g ⇒ uτ = 0, |(σn)τ | = g ⇒ uτ 6= 0 , − (σn)τ = g uτ |uτ |    on S. (1.9) As pointed out by C. Leroux [41], such a theory can represent only a limited range of possible situations. The purpose of this chapter is to numerically analyze by means of finite element approximation equations (1.1)–(1.5), and (1.2)–(1.5),(1.8). At this juncture, it is important to recall that this type of nonlinear slip boundary conditions as far as fluid flows are concerned was first introduced by Fujita in [23, 24]. This is in continuation of a series of investigations aimed at the analysis of Stokes and Navier-Stokes equations driven by nonlinear slip boundary conditions of friction type (see [75, 76, 49]). In order to provide a background for a better mathematical understanding of the problems, we shall introduce in Section 2.2 some needed tools, and quickly indicate how the problems are solvable. At this step, we recall that in C. Leroux and Tani [41, 42] a fixed point argument is used to establish the solvability of a class of problems similar to what we want to study. It is re-introduced here because of its usefulness in the finite element analysis. Hence one can see a sort of “continuum” between the continuous and discrete analysis. The finite element formulations for both Stokes and Navier- Stokes equations are derived in Section 1.3. The finite elements are defined on conforming triangular mesh as introduced in [62], and in each triangle the velocity and pressure are taken so that the Babuska-Brezzi’s condition [79, 80] is satisfied. Here, we do not use penalty method, or pressure stabilized method to enforce the incompressibility condition. Instead we use a direct method and sufficient conditions of existence of solutions are employed to derive a priori error estimates in Section 1.3. In Section 1.4, Uzawa’s algorithm is formulated and analyzed for solving the Stokes and Navier-Stokes finite element discretization. It is shown that the Uzawa’s algorithm converges. In Section 1.5 numerical simulations that confirm the predictions of the theory are exhibited. where ∂|·| is the sub-differential of the real valued function |·| with |w| 2 = w ·w. It should be mentioned that different boundary conditions describe different phys- ical phenomena. The slip boundary conditions of friction type (2.5) can be justi- fied by the fact that frictional effects of the fluid at the pores of the solid can be very important. The class of boundary condition (2.5) was introduced by Fujita in [23], where he studied some hydrodynamics problems, such as the blood flow in a vein of an arterial sclerosis patient and the avalanche of water and rocks. Subsequently, many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes and Brinkman- Forchheimer equations under such boundaries condition. Details can be found in [41, 23, 28, 40, 42, 87, 31, 27, 29, 88, 89, 24, 32, 90] among others. But the combination of (2.5) with the p-Laplacian has not yet been considered in the literature, and in this work we give a detailed mathematical analysis on the ex- istence and uniqueness of weak solution. The aim of this study is to contribute to the numerical analysis of flows problem driven by non-conventional boundary conditions. Hence, our main focus is to analyze numerically (2.1)—(2.5) via finite element approximations. That is to establish the convergence of the finite element solution. It is manifest that (2.1)—(2.5) has many numerical challenges among others; the nonlinear operator (p-Laplacian), the incompressible condition and the related pressure, and the nontrivial boundary condition (2.5) which brings a non-differentiable expression into the variational formulation of the problem. Hence, our second contribution here is to formulate and analyze an algorithm well adapted and easy to implement for the numerical challenges mentioned. Even though many researches have been done for the approximations of varia- tional inequalities [91, 63, 48, 72, 71] (just to mention a few), not much research in theoretical numerical analysis has been done for the kind of problem described by (2.1)—(2.5). Li and Li [35] proposed a penalty finite element approximation method for the Stokes equation with nonlinear slip boundary conditions (2.5). They proved the optimal order error estimate provided that the velocity is H2 up to the boundary, however, no numerical simulations are exhibited. An and Li [78] proposed a penalty finite element method for the steady Navier-Stokes equations. The Mathematical analysis of this chapter borrows heavily on the contribution of Reddy [70] and Han and Reddy [48], where sufficient conditions for existence and uniqueness are derived for the kind of weak formulations we analyze here, while the solution procedure we propose is divided in three steps. The first step is based on some works of R. Glowinski [91] in that, we associated to a steady problem an evolution problem in which only the long time effect is taken into consideration. Next, because of the incompressibility condition, and the non differentiable term appearing in the variational problem to solve, we approximated then the problem by a sequence of penalized/regularized “better behaved” variational equations and justify the approximations by some convergence results. Thirdly, to improve the performance of our scheme, we add to the problem obtained in step 2 a viscos- ity term and show that the new “perturbed” problem converges to the original variational formulation. All these theoretical results are supported by numerical simulations indicating the robustness of our algorithm. The rest of this chapter is organized as follows. We give some notations, formulate the variational models in section 2.2 and indicate how existence of weak solution is obtained. In section 2.3, we formulate the finite element procedure, explain how existence and uniqueness of solution is obtained and derive error estimates. Section 2.4 is concerned with the algorithm, while section 2.5 deals with numerical simulations.

Numerical Algorithm

In this section, we formulate and analyze the algorithm for the implementation of (2.16). We first regularize the formulation (2.16) by replacing the non differ- entiable functional j by a “better behaved” approximation jε where ε is a small positive parameter (see B.D. Reddy [70]) . It should be mentioned that the in- troduction of the new functional jε transforms the variational inequality problem into a variational equation. The next step in our strategy consists of eliminating the incompressibility constraint by penalizing the regularized problem by adding a coercive-like term in the form η(p, q), of course η is a small positive parameter. We recall that the transformed problem is very close to the original one in the sense that when ε, η tend to zero, one recovers the original problem. Finally, the perturbed problem with parameters is solved by considering the numerical solu- tion of the long time behavior of an appropriate initial value problem in Vh ×Mh. One of the advantages of using this approach is that a linear scheme can be for- mulated for a nonlinear problem. We next present the details of our approach. The non-differentiable functional j is replaced in (2.16) by the regularized func- tional jε defined by jε(v) = Z S g p |v| 2 + ε 2 ds. (2.44) Note that jε satisfies the following properties: (i) jε is convex and differentiable, with Gateaux derivative and

Contents :

• Declaration
• Acknowledgements
• Abstract
  • Introduction
  • 0.1 Thesis overview and our contributions
  • 0.2 Generalities on variational inequality and finite element approximation
    • 0.2.1 Function spaces
    • 0.2.2 Elements of nonlinear analysis
    • 0.2.3 Standard results on variational inequalities
    • 0.2.4 Preliminaries on finite element approximations
  • 1 Finite element analysis on steady Navier-Stokes and Stokes equations driven by threshold slip boundary conditions
  • 1.1 Introduction
  • 1.2 Preliminaries and Variational Formulations
    • 1.2.1 Notations and Preliminaries
    • 1.2.1.1 Mixed Variational formulation of (1.1)–(1.5)
    • 1.2.1.2 Mixed Variational formulation (1.2)–(1.5) and (1.8)
  • 1.3 Finite element approximations
    • 1.3.1 Finite element approximation of the variational inequality (1.20)
    • 1.3.1.1 Existence and uniqueness of solution
    • 1.3.1.2 A priori error estimate
    • 1.3.2 Finite element approximation of the variational inequality (1.30)
    • 1.3.2.1 Existence and uniqueness of solution
    • 1.3.2.2 A priori error estimate
  • 1.4 Numerical Algorithm
  • 1.4.1 Numerical algorithm for Stokes variational inequality (1.39)
  • 1.4.2 Numerical algorithm for Navier-Stokes variational inequality (1.56)
  • 1.5 Numerical experiments
    • 1.5.1 Numerical examples for Stokes problem (1.1)-(1.5)
    • 1.5.2 Numerical examples for Navier-Stokes problem (1.2)–(1.5),(1.8)
    • 1.5.3 Numerical accuracy check
  • 2 Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions
  • 2.1 Introduction
  • 2.2 Variational Formulations
    • 2.2.1 Notation
    • 2.2.2 Mixed variational formulation
  • 2.3 Finite element approximation of the variational inequality (2.7)
    • 2.3.1 Preliminaries and existence of solution
    • 2.3.2 A priori error estimate
    • 2.3.3 Rate of convergence
  • 2.4 Numerical Algorithm
  • 2.5 Numerical experiments
    • 2.5.1 Numerical accuracy check
    • 2.5.2 Driven cavity
  • 3 On the long-time stability of the Crank-Nicolson scheme for the 2D Navier-Stokes equations driven by threshold slip boundary conditions
  • 3.1 Introduction
  • 3.2 Preliminaries and Variational formulation
  • 3.3 Numerical scheme
  • 3.4 The (V h, k · kh)− stability
  • 3.5 The (V h, k · k1,h)- stability
  • Conclusion
  • Bibliography

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